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\author{Guojun Zhu}
\title{Narrow Feshbach Resonance}
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\section{}
 The full two-channel hamiltonian is
\begin{equation}\label{eq:uvw:hamiltonian}
\begin{split}
 H=&\sum_\vk\epsilon^a_\vk{}a^+_\vk{}a^{}_\vk+\sum_\vk\epsilon^b_\vk{}b^+_\vk{}b^{}_\vk+\sum_\vk\epsilon^c_\vk{}c^+_\vk{}c^{}_\vk\\
  &+\nth{2}\sum_{\vk\vk'}U_{\vk\vk'}a^+_\vk{}b^+_{-\vk}{}b^{}_{-\vk'}a^{}_{\vk'}
	+\nth{2}\sum_{\vk\vk'}V_{\vk\vk'}a^+_\vk{}c^+_{-\vk}{}c^{}_{-\vk'}a^{}_{\vk'}\\
 &+\nth{2}\sum_{\vk\vk'}Y_{\vk\vk'}a^+_\vk{}b^+_{-\vk}{}c^{}_{-\vk'}a^{}_{\vk'}
	+\nth{2}\sum_{\vk\vk'}Y^*_{\vk\vk'}a^+_{\vk'}{}c^+_{-\vk'}{}b^{}_{-\vk}a^{}_{\vk}
\end{split} 
\end{equation}
 We start from the ansatz as 
\begin{equation}\label{eq:ansatz}
 \ket{\Psi}=\prod_\vk\br{u_\vk+v_\vk{}a^\dg_\vk{}b^\dg_{-\vk}+w_\vk{}a^\dg_\vk{}c^\dg_{-\vk}}\ket{0}
\end{equation}
The free energy is (neglect Hatree terms as they can be absorbed into chemical potential)
\begin{equation}\label{eq:uvw:F}
 \begin{split}
  &F\equiv\av{H-\mu{}N}\\
    =&\sum(\xi^{ab}_\vk)\abs{v_\vk}^2+\nth2\sum_{\vk\neq\vk'}U_{\vk\vk'}v^{}_{\vk'}u^*_{\vk'}u^{}_\vk{}v^*_\vk\\
    &+\sum(\xi^{ac}_\vk)\abs{w_\vk}^2
      +\nth2\sum_{\vk\neq\vk'}V_{\vk\vk'}w^{}_{\vk'}u^*_{\vk'}u^{}_\vk{}w^*_\vk\\
    &  +\nth2\sum_{\vk\neq\vk'}Y_{\vk\vk'}w^{}_{\vk'}{u^{*}_{\vk'}}v^*_\vk{}u^{}_\vk
        +\nth2\sum_{\vk\neq\vk'}Y^*_{\vk\vk'}w^*_{\vk}{u^{}_{\vk}}v^{}_{\vk'}{}u^{*}_{\vk'}
 \end{split}
\end{equation}
Where 
\begin{equation*}
 \xi^{ab}_\vk=\epsilon^a_\vk+\epsilon^b_\vk-2\mu,\qquad
  \xi^{ac}_\vk=\epsilon^a_\vk+\epsilon^c_\vk-2\mu
 \end{equation*}



Introduce the new parameter $F_{\vk}$ and $G_{\vk}$,  solve $u_{\vk}$, $v_{\vk}$, $w_{\vk}$ with $F_{\vk}$ and $G_{\vk}$ (treat everything as real)
\begin{gather}
u_{\vk}^2+v^{2}_{\vk}+w^{2}_{\vk}=1\\
u_{\vk}v_{\vk}=F_{\vk}\\
u_{\vk}w_{\vk}=G_{\vk}
\end{gather}
One complication is that $u_\vk$ is a monotonic function of $k$ while $F_\vk$ is not in BCS/BCS-like situation.  So we need to be careful when take the square root.  We introduce $\sgn_k$ for such purpose.  \footnote{\label{foot:20100909:sgn} $\sgn_k=1$  for the whole region of BEC case, or for $k$ is large in BCS case. In BCS case, when $k$ is small, $\sgn_{k}=-1$.  In single-channel case $\sgn_{k}=\sgn(\epsilon_{k}-\mu)$, the turning point is at chemical potential $\mu$.  However, in two-channel, this is more delicate.  The turning point is close to chemical potenial, but with a shift. It is $\sgn_k=\sgn(\epsilon^{ab}_{\vk}-2\mu+  G_{\vk}^2\eta)$.  This is very important in \eef {eq:20100909:number}}\begin{equation}
\begin{split}
u_{\vk}^2=&\frac{1}{2} \left(1+\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)\\
v^{2}_{\vk}=&\frac{2 F_{\vk}^2}{1+\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}
=\frac{ F_{\vk}^2}{2( F_{\vk}^2+ G_{\vk}^2)} \left(1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)\\\
w^{2}_{\vk}=&\frac{2 G_{\vk}^2}{1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}
=\frac{G_{\vk}^2}{2( F_{\vk}^2+ G_{\vk}^2)} \left(1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)
\end{split}
\end{equation}
Below, we adopt $\sgn_k=1$. (This works for BEC and most part of BCS; and in final equation, the sign function is taken care of automatically.  It is easy to verify we arrive at the same number equation in the other case where $\sgn_{k}=-1$. ) Derivatives over $F_{\vk}$ are
\begin{equation}
\begin{split}
\pdiff{u_{\vk}^2}{F_\vk}=&-\frac{2 F_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}\\
\pdiff{v_{\vk}^2}{F_\vk}=&\frac{2 F_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}-\frac{8 F_{\vk} G_{\vk}^2}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}\\
\pdiff{w_{\vk}^2}{F_\vk}=&\frac{8 F_{\vk} G_{\vk}^2}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}
=\frac{F_{\vk} G_{\vk}^2 \left(1-\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}{2\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} (F_{\vk}^2 + G_{\vk}^2)^2}
\end{split}
\end{equation}
Similarly, the derivative over $G_{\vk}$ can be obtained by exchange $F_{\vk}$ ($v_{\vk}$) and $G_{\vk}$ ($w_{\vk}$).
\begin{equation}
\begin{split}
\pdiff{u_{\vk}^2}{G_\vk}=&-\frac{2 G_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}\\
\pdiff{v_{\vk}^2}{G_\vk}=&\frac{8 F_{\vk} ^{2}G_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}
=\frac{F_{\vk}^{2}G_{\vk} \left(1-\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}{2\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} (F_{\vk}^2 + G_{\vk}^2)^2}\\
\pdiff{w_{\vk}^2}{G_\vk}=&\frac{2 G_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}}-\frac{8 F_{\vk} ^{2}G_\vk}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}
\end{split}
\end{equation}
\begin{figure}[hhtb]
	\centering
		\includegraphics[width=.50\textwidth]{image/FeshbachPotential}
	\caption{Feshbach Resonance Potential\label{fig:FeshbachPotential}}	
\end{figure}

The gap equations are 
\begin{subequations}\label{eq:20100909:fullgap}
\begin{gather}
\frac{2F_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}} \xi^{ab}_{\vk}+\frac{8 F_{\vk} G_{\vk}^2}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}\eta+U_{\vk\vk'}F_{\vk'}+Y_{\vk\vk'}G_{\vk'}=0
\label{eq:20100909:fullgapa}\\
\frac{2G_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}} \xi^{ac}_{\vk}-\frac{8 F_{\vk}^{2} G_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2} \left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2}\eta+V_{\vk\vk'}G_{\vk'}+Y_{\vk\vk'}F_{\vk'}=0
\label{eq:20100909:fullgapb}
\end{gather}
\end{subequations}
where $\eta=\epsilon^{ac}_{\vk}-\epsilon^{ab}_{\vk}$ is the bare Zeeman energy difference and is large than most energy scale, such as  $E_{F}$.  It should be in the order of binding energy of the close-channel bound state.   

The number equation is (see footnote (\ref{foot:20100909:sgn}) for $\sgn_{k}$)
\begin{equation}\label{eq:20100909:number}
N=\sum_{\vk}(v_{\vk}^{2}+w_{\vk}^{2})=\sum_{\vk}\frac{1}{2} \left(1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)
\end{equation} 


\subsection{Approximate close-channel with bound-state level}
Let us look at \eef{eq:20100909:fullgapb}.  It is energetically very disadvantageous to deviate from the bound state as the binding energy is much  larger than Fermi energy, i.e. the bound-state is relatively small in real space comparing to inter-particle distance.     This equation must close to  the solution for the isolated close-channel \sch.  
\begin{equation}\label{eq:20100915:twobody}
{\phi^{0}_{\vk}}(\epsilon^{\text{pair}}_{\vk})+\nth{2}V_{\vk\vk'}\phi^{0}_{\vk'}=-E^{0}\phi^{0}_{\vk'}
\end{equation}
This equation relates region (in k-space) much larger than $k_{F}$.  On the other hand, the open-channel 
$F_{\vk}$ is only substantial in region around or not too high above  $k_{F}$.  Therefore, as the first order approximation, we takes $F_{\vk}=0$ for this equation. So we can drop the second term and the denominator in the first term ($G_{\vk}\ll1$ all the time) of \eef{eq:20100909:fullgapb}\footnote{ This is certainly OK for he BEC end where $F_{k}\ll1$ all the time.  It is less satisfactory, might be problematic,  when $F_{k}$ is close to maximum $\nth2$, this happens around Fermi energy in BCS limit, but the approximation is still OK for other places. So at least for bulk of the region of $G_{\vk}$ satisfies \eef{eq:20100915:gapb}}.  In  open-channel equation, \eef{eq:20100909:fullgapa}, we drop the factor $\left(1+\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)^2$ in the second term for easier calculation, (no strong reason yet, but this term can only takes value $[1,4]$, and the second term is relatively minor comparing to the first). We write down the approximated gap equations
\begin{subequations}\label{eq:20100915:gap}
\begin{gather}\label{eq:20100915:gapa}
\frac{2F_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}} (\epsilon^{ab}_{\vk}-2\mu+  G_{\vk}^2\eta)+U_{\vk\vk'}F_{\vk'}+Y_{\vk\vk'}G_{\vk'}=0\\
\label{eq:20100915:gapb}
{2G_{\vk}}(\epsilon^{ab}_{\vk}-2\mu+\eta)+V_{\vk\vk'}G_{\vk'}+Y_{\vk\vk'}F_{\vk'}=0
\end{gather}
\end{subequations}
Here we express $\xi^{ab}_{\vk}=\epsilon^{ab}_{\vk}-2\mu$, $\epsilon^{ac}_{\vk}=\epsilon^{ab}_{\vk}+\eta$.  And many-body close-channel wave function $G_{\vk}$ is proportional to two-body bound-state wave function $\phi_{\vk}^{0}$,  $G_{\vk}=\alpha\phi^{0}_{\vk}$. 
\eef{eq:20100915:gapb} becomes
\begin{equation}\label{eq:20100915:GF}
G_{\vk}=\frac{Y_{\vk\vk'}F_{\vk'}}{2(E^{0}-\eta+2\mu)}
\end{equation}
Plug this back into the last term of \eef{eq:20100915:gapa}, we have 
\begin{equation}
\frac{2F_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}} (\xi^{ab}_{\vk}+  G_{\vk}^2\eta)+U_{\vk\vk'}F_{\vk'}+\frac{Y_{\vk\vk'}Y_{\vk'\vk''}}{2(E^{0}-\eta+2\mu)}F_{\vk''}=0
\end{equation}
Now considering the last two terms has weak dependecy in low k, we can set 
\begin{equation}\label{eq:20100915:gap1}
\frac{2F_{\vk}}{\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}} (\xi^{ab}_{\vk}+  G_{\vk}^2\eta)\equiv\Delta_{\vk}=-(U_{\vk\vk'}F_{\vk'}+\frac{Y_{\vk\vk'}Y_{\vk'\vk''}}{2(E^{0}-\eta+2\mu)}F_{\vk''})
\end{equation}
We can express $F_{\vk}$ according to $\Delta_{\vk}$,  (ignore the higher order of $G_{\vk}$)
\begin{equation}\label{eq:20100915:F}
F_{\vk}=\frac{\Delta_{\vk}}2\sqrt{\frac{(1-4G_{\vk}^{2})}{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}
\end{equation}
Put it back into the second half of gap equation (\ref{eq:20100915:gap1}), 
\begin{equation}\label{eq:20100915:onechannel}
\Delta_{\vk}=-\sum_{\vk'}\br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)}}\frac{\Delta_{\vk'}}2\sqrt{\frac{(1-4G_{\vk'}^{2})}{(\xi^{ab}_{\vk'}+  G_{\vk'}^2\eta)^{2}+\Delta_{\vk'}^{2}}}
\end{equation}
\subsection{Renormalization of gap equation}
At low k, $\Delta_{\vk}$ has weak dependency on k and we can take $\Delta_{\vk'}$ out of the summation,  then go through the same procedure as in \cite{Leggett,Fetter} to renormalize the equation. There are more than one options to renormalize gap equation \eef{eq:20100915:onechannel}.  The part that needs to be renormalized out is  high energy summation of $\sqrt{\frac{(1-4G_{\vk'}^{2})}{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}$, it approaches $\nth{\epsilon_{\vk}}$ in high energy.  Several sightly different physical quantities have the summation of the same high energy limit.  
\begin{enumerate}

\item We can also notice that $G_{\vk}\rightarrow0$ at high energy.  So we can simply takes the normal \emph{zero-energy} T-matrix with detuning related to chemical potential.  
\begin{equation}\label{eq:20101004:renormGap1}
\nth{{t_{0}}(\mu)}=\sum_{\vk}
\br{\nth{\epsilon_{\vk}}-\frac{\sqrt{(1-4G_{\vk}^{2})}}{\sqrt{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}}
\end{equation} 
\begin{gather}
{t_{0}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\label{eq:20101004:t01}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+\mu)}}\label{eq:20101004:tu1}\\
{K}=\nth{\epsilon_{\vk}}\delta_{\vk\vk'}\label{eq:20101004:tk1}
\end{gather}
Here Eqs. (\ref{eq:20101004:t01}-\ref{eq:20101004:tk1}) follows the same two-body formula for zero-energy T-matrix element.  However, the detuning is shifted by a many-body quantity $\mu$ that should be determined by solving gap equation with number equation.    
\item  Alternatively, we notice that $\xi_{\vk}=\epsilon_{\vk}-\mu$, high energy limit can also be written as 
$\nth{\epsilon_{\vk}-\mu}$.  This leads to the T-matrix at energy $\mu$, the same detuning as before.  
\begin{equation}\label{eq:20101004:renormGap2}
\nth{{t_{\mu}}(\mu)}=\sum_{\vk}
\br{\nth{\epsilon_{\vk}-\mu}-\frac{\sqrt{(1-4G_{\vk}^{2})}}{\sqrt{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}}
\end{equation} 
\begin{gather}
{t_{\mu}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\label{eq:20101004:t02}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+\mu)}}\label{eq:20101004:tu2}\\
{K}=\nth{\epsilon_{\vk}-\mu}\delta_{\vk\vk'}\label{eq:20101004:tk2}
\end{gather}
The advantage of this is that introduce the effective range $r_{0}$ for finite energy T-matrix. 
\end{enumerate}


It seems the first method would be the most convenient choice.  

\subsection{Two effects of narrow resonance}
There are two possibilities for narrow resonance:  
\begin{enumerate}
\item Pauli exclusion between two channels only in three species, and only important where the close-channel weight is large.  
\item Fermi energy is larger than the width that scattering amplitude (at finite energy, NOT zero-energy value simply proportional to $a_{s}$) changes the sign.  Therefore $a_{s}$ is no longer a good indicator for the interaction.  However, it might still be OK to take $a_{s}$ simply as the quantity introduced by renormalization.   This has been explored by several papers already \cite{NarrowJensen1,NarrowJensen,GurarieNarrow}.  
\end{enumerate}
Four species case can fall into narrow resonance in  the second sense.  \cite{NarrowJensen1,NarrowJensen} uses the single-channel approach but a T-matrix with effective range $r_{0}$, the narrow resonance is $r_{0}k_{F}\ll1$.  In \cite{NarrowJensen}, a single-channel bare interaction is assumed and (unnormalized) gap, number equations  are  solved numerically with different interaction parameters corresponding to different effective ranges. The result is some small corrections in gap, chemical potential...   It is not a two-channel treatment.  

\cite{GurarieNarrow} treats the problem in two-channel (molecule+fermi).  The criteria for narrowness is similar.  


\subsection{Narrow Resonance in the Second Sense}
In narrow resonance, Fermi sea is deep and only part of the fermi sea (strongly) interacts/resonants with the close-cannel bound-state.   \index{Feshbach Resonance!Narrow}
\begin{figure}[hhtb]
	\centering
	         \subfloat[$\delta>E_{F}$]{\label{fig:narrowFR:aboveSea}\includegraphics[width=.20\textwidth]{image/narrowFR2.eps}}\quad
		\subfloat[$E_{F}>\delta>0$]{\includegraphics[width=.30\textwidth]{image/narrowFR.eps}\label{fig:narrowFR:inSea}}\quad
		\subfloat[$\delta<0$]{\label{fig:narrowFR:belowSea}\includegraphics[width=.20\textwidth]{image/narrowFR3.eps}}
	{\caption{Narrow Resonance\label{fig:narrowFR}}
	\parbox{0.7\textwidth}{\small{In fact, in Fig. \subref{fig:narrowFR:inSea} chemical potential would be close to the close-channel bound state level (besides small shift due to open-channel intra-channel coupling) and ``Fermi sea'' above is likely empty. }}}
\end{figure}
Fig. (\ref{fig:narrowFR}).  Here the zero-energy T-matrix is less  physically relevant than those with finite $T(E_{F}-\delta)$ (finite energy measured from Fermi surface instead 0).  The more commonly used s-wave scattering length $T(E)$ likely changes sign within Fermi sea.  $T(E)$ dose not depends on $a_{s}$, but also on effective range $r_{0}$.  

Here BCS/BEC extremes are subtle in the meaning.  If that refers to close-channel bound-state above or below Fermi sea and the interaction is necessarily small as the condition for narrow resonance, they are not very interesting by definition.  When the close-channel bound-state is below zero, most fermions are in close-channel bound-state.  It is BEC of molecules and molecules are mostly in close-channel. When the level is above the Fermi sea, it is fairly far from zero and actually only open-channel fermions near Fermi surface feel relatively strong attraction and they are in BCS.  The more interesting part where it sits in the Fermi sea is always the  case in middle.  

\subsection{Chemical potential}
\emph{Chemical potential is determined in different ways between narrow or broad resonance.  }In broad case, it is determined mostly by open-channel \eef{eq:20100915:gapa}; in narrow case, chemical potential is determined except the very BCS end,  by where the close-channel bound-state level sits relative to Fermi sea.  In the extreme narrow case (without open-channel interaction as \cite{GurarieNarrow}), the level is exactly where chemical potential sits, cutting Fermi sea, depleting everything above in open-channel and putting them into close-channel.   

By putting the chemical potential $\mu$ in the proper position, we can see the narrow/broad resonance even in the four species case.  In Eq. (\ref{eq:20100915:t0}-\ref{eq:20100915:tk}), there are two many-body effects: $\sqrt{1-4G_{\vk}^{2}}$ from three-species Pauli exclusion; chemical potential in detuning term $E^{0}-\eta+\mu$, which is common in either three or four-species case.  And the later reduces to $\mu=0$ \footnote{In BEC side, $\mu<0$ and is controlled by mostly two-body attraction.  But that is not proper for real two-body limit, which should be $\mu=0$.}in zero-density which is two-body case. 

Imaging we start fairly far away from resonance (BCS side, $\delta=\eta-E^{0}>0$), and increase the density, in the beginning, $\mu$ is negligible, and inter-channel coupling term $\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+\mu)} $ is small; as $\mu$  increases, $-(\delta-\mu)$ gets closer and closer to zero, and this terms increases until the part of the Fermi sea gets into resonance.  Then most of adding fermions go into close-channel. 






\subsection{More on chemical potential shift for detuning}
Let us image a sweep of $\delta$ from BCS end.  At the very BCS end, $\delta$ is positive and large than $\mu\approx{}E_{F}$ (Fig. \ref{fig:narrowFR:aboveSea}). Resonant term in interaction is relatively small and close-channel weight is small too.  We are very well in the  weak-attraction (slightly enhanced by resonance) BCS-like state in open-channel.  However, the detuning is shifted by $\mu\approx{}E_{F}$, so the resonance is reached earlier, and the larger the density, the earlier it does.  

As the detuning gets close to Fermi surface, chemical potential decreases from $E_{F}$. For narrow resonance, $E_{F}$ is large than the resonance energy scale.  The resonance is reached probably before detuning reaches Fermi surface.  If we ignore the shift due to open-channel intra-channel coupling for the moment, the resonance is very close at the point where $\delta=E_{F}$.  After $\delta$ drops into Fermi sea (Fig. \ref{fig:narrowFR:inSea}), chemical potential $\mu$ tracks the close-channel bound-state closely and the denominator of resonant term $-(\delta-\mu)$ keeps very small, and in some sense, the Feshbarch resonance is enhanced by the many-body  physics.  Open-channel still has energy advantages below $\mu$, so both channels are important.  

After $\delta$ drops below zero (Fig. \ref{fig:narrowFR:belowSea}), chemical potential still tracks $\delta$, but now most weight is in close-channel and the bound-state is mostly made of close-channel.  


\section{Number/Gap equations}
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